Recent and promising advancements in fields ranging from materials science to quantum physics are now being used to generate new quantum-system-based technologies. These quantum systems can be used to encode and transmit quantum information. In particular, quantum systems comprising just two discrete states, represented by “|0” and “|,” can potentially be employed in a variety of quantum-system-based applications including quantum information encoding and processing, optical quantum lithography, and metrology, just to name a few. A quantum system comprising two discrete states is called a “qubit system,” and the states |0 and |1, called “qubit basis states,” can also be represented in set notation as {|1, |1}. A qubit system can exist in the state |0, the state |1, or in any of an infinite number of states that simultaneously comprise both |0 and |1, which can be mathematically represented by a linear superposition of states as follows:|ψ=α|0+β|1The state |ψ is called a “qubit,” and the parameters α and β are complex-valued coefficients satisfying the condition:|α|2+|β|2=1
Performing a measurement on a quantum system is mathematically equivalent to projecting the state of the quantum system onto one of the basis states, and, in general, the probability of projecting the state of the quantum system onto a basis state is equal to the square of the coefficient associated with the basis state. For example, when the state |ψ of the qubit system is measured in the basis {|0, |1}, one has a probability |α|2 of finding the quantum system in the state |0 and a probability |β|2 of finding the quantum system in the state |1.
The infinite number of pure states associated with a qubit system can be geometrically represented by a unit-radius, three-dimensional sphere called a “Bloch sphere”:
          ψ    〉    =                    cos        ⁡                  (                      θ            2                    )                    ⁢                      0        〉              +                  ⅇ                  ⅈ          ⁢                                          ⁢          ϕ                    ⁢              sin        ⁡                  (                      θ            2                    )                    ⁢                      1        〉            where
0≦θ<π, and
0≦φ<2π.
FIG. 1A illustrates a Bloch sphere representation of a qubit system. In FIG. 1A, lines 101-103 are orthogonal x, y, and z Cartesian coordinate axes, respectively, and a Bloch sphere 106 is centered at the origin. There are an infinite number of points on the Bloch sphere 106, each point representing a unique state of a qubit system. For example, a point 108 on the Bloch sphere 106 represents a unique state of a qubit system that simultaneously comprises, in part, the state |0> and, in part, the state |1. However, once the state of the qubit system is measured in the basis {|0, |1}, the state of the qubit system is projected onto the state |0 110 or onto the state |1 112.
Photon states of electromagnetic radiation can be used as qubit basis states in quantum information processing and quantum computing applications. The term “photon” refers to a single quantum of excitation energy of an electromagnetic field mode of electromagnetic radiation. The electromagnetic radiation can be in the form of propagating electromagnetic waves, each electromagnetic wave comprising both a transverse electric field component, {right arrow over (E)}, and an orthogonal transverse magnetic field component, {right arrow over (B)}. FIG. 1B illustrates the transverse electric and magnetic field components of an electromagnetic wave propagating in the direction, {right arrow over (k)}. As shown in FIG. 1B, the electromagnetic wave is directed along the z-axis 120. The transverse electric field (“TE”) component {right arrow over (E)} 122 and the transverse magnetic field (“TM”) component {right arrow over (B)} 124 are directed along the orthogonal x- and y-axes 126 and 128, respectively. Although the TE and TM are shown in FIG. 1B to have identical amplitudes, in real life the amplitude of the TM component is smaller than the amplitude of the TE component by a factor of 1/c, where c represents the speed of light in free space (c=3.0×108 m/sec). Because of the large discrepancy in the magnitude of the electric field component and the magnitude of the magnetic field component, the electric field component alone typically accounts for most of the electromagnetic wave interactions with matter.
Polarized photon states of electromagnetic waves can also be used as qubit basis states in quantum information processing and quantum computing. Two commonly used basis states are vertically and horizontally polarized photons of electromagnetic waves. The terms “vertical” and “horizontal” are relative with respect to a coordinate system and are used to refer to electromagnetic waves that are oriented orthogonal to one another. FIGS. 2A-2B illustrates vertically and horizontally polarized photons, respectively. In FIGS. 2A-2B, vertically and horizontally polarized photons are represented by oscillating, continuous sinusoidal waves that represent the electric field components propagating along z-coordinate axes 202 and 204, respectively. As shown in FIG. 2A, a vertically polarized photon |V corresponds to an electric field component that oscillates in the yz-plane. Directional arrow 206 represents one complete oscillatory cycle of the electric field component of |V in the xy-plane 208 as |V advances along the z-coordinate axis 202 through one complete wavelength. In FIG. 2B, a horizontally polarized photon |H corresponds to an electric field component that oscillates in the xz-plane. Directional arrow 210 represents one complete oscillatory cycle of the electric field component of |H in the xy-plane 212 as |H advances along the z-coordinate axis 204 through one complete wavelength.
The state of a system comprising two or more qubit systems can be represented by a tensor product of qubits, each qubit associated with one of the qubit systems. For example, the tensor product of a system comprising a first qubit system and a second qubit system is given by:|ψ12=|ψ|ω2 where the state of the first qubit system is:
                  ψ      〉        1    =            1              2              ⁢          (                                                0            〉                    1                +                                          1            〉                    1                    )      and the state of the second qubit system is:
                  ψ      〉        2    =            1              2              ⁢          (                                                0            〉                    2                +                                          1            〉                    2                    )      The state |ψ12 can also be rewritten as a linear superposition of products of basis states:
                  ψ      〉        12    =                                        ψ          〉                1            ⁢                                  ψ          〉                2              =                  1        2            ⁢              (                                                                            0                〉                            1                        ⁢                                                          0                〉                            2                                +                                                                    0                〉                            1                        ⁢                                                          1                〉                            2                                +                                                                    1                〉                            1                        ⁢                                                          0                〉                            2                                +                                                                    1                〉                            1                        ⁢                                                          1                〉                            2                                      )            where the terms |01|02, |01|12, |11|02, and |1|12 are a basis of the tensor product space. Each product state in the state |ψ12 has an associated coefficient of ½, which indicates that when the state of the first qubit system is measured in the bases {|01,|11}, and the state of the second qubit system is measured in the basis {|02,|12} there is a ¼ (|½|2) probability of the combined qubit systems being found in any one of the product states.
Certain states of the combined qubit systems, however, cannot be represented by a product of associated qubits. These qubit systems are said to be “entangled.” Quantum entanglement is a unique property of quantum mechanics in which the states of two or more quantum systems are correlated, even though the quantum systems can be spatially separated. An example entangled-state representation of an entangled two-qubit system is given by:
                          ψ        +            〉        12    =            1              2              ⁢          (                                                                0              〉                        1                    ⁢                                                  1              〉                        2                          +                                                          1              〉                        1                    ⁢                                                  0              〉                        2                              )      The entangled state |ψ+12 cannot be factored into a product of the qubits α1|01+β1|11 and α2|02+β2|12, for any choice of the parameters α1, β1, α2, and β2.
The state of an un-entangled, two-qubit system can be distinguished from the state of an entangled, two-qubit system as follows. Consider an un-entangled, two-qubit system in the state |ψ12. Suppose a measurement performed on the first qubit system in the basis {|01, |11} projects the state of the first qubit system onto the state |01. According to the state |ψ12, the state of the un-entangled, two-qubit system immediately after the measurement is the linear superposition of states (|01|02+|01|12)/√{square root over (2)}. When a second measurement is performed on the second qubit system in the basis {|2, |12} immediately following the first measurement in an identical reference frame, there is a ½ probability of projecting the state of the second qubit system onto the state |02 and a ½ probability of projecting the state of the second qubit system onto the state |12. In other words, the state of the second qubit system is not correlated with the state of the first qubit system.
In contrast, consider an entangled, two-qubit system in the entangled state |ψ+12. Suppose that a first measurement performed on the first qubit system in the basis {|01,|11} also projects the state of the first qubit system onto the state |01. According to the entangled state |ψ+2, the state of the entangled, two-qubit system after the first measurement is the product state |01|12. When a second measurement is performed on the second qubit system in the basis {|02,|12}, the state of the second qubit system is |12 with certainty. In other words, the state of the first qubit system is correlated with the state of the second qubit system.
Entangled quantum systems have a number of different and practical applications in fields ranging from quantum computing to quantum information processing. In particular, the polarization entangled-photons described above can be used in quantum information processing, quantum cryptography, teleportation, and linear optics quantum computing. Examples of polarization entangled-photons that can be used in a number of different entangled-state applications are the Bell states given by:
                                                                    ψ              -                        〉                    =                                    1                              2                                      ⁢                          (                                                                                                                H                      〉                                        1                                    ⁢                                                                                  V                      〉                                        2                                                  -                                                                                                  V                      〉                                        1                                    ⁢                                                                                  H                      〉                                        2                                                              )                                      ,                                                                                ψ              +                        〉                    =                                    1                              2                                      ⁢                          (                                                                                                                H                      〉                                        1                                    ⁢                                                                                  V                      〉                                        2                                                  -                                                                                                  V                      〉                                        1                                    ⁢                                                                                  H                      〉                                        2                                                              )                                      ,                                                                                ϕ              -                        〉                    =                                    1                              2                                      ⁢                          (                                                                                                                V                      〉                                        1                                    ⁢                                                                                  V                      〉                                        2                                                  -                                                                                                  H                      〉                                        1                                    ⁢                                                                                  H                      〉                                        2                                                              )                                      ,                                  ⁢        and                                                                ϕ            +                    〉                =                              1                          2                                ⁢                      (                                                                                                    V                    〉                                    1                                ⁢                                                                          V                    〉                                    2                                            +                                                                                        H                    〉                                    1                                ⁢                                                                          H                    〉                                    2                                                      )                              where the subscripts “1” and “2” can represent different transmission channels or different wavelengths.
Although polarization-entangled photons have a number of potentially useful applications, polarization-entangled photon sources typically cannot be practically implemented in a wide variety of entangled state applications. For example, in “New High-Intensity Source of Polarization-Entangled Photon Pairs,” by Kwiat et al., Physical Review Letters, vol. 75, 4337, (1995), Kwiat describes a high-intensity source of polarization entangled-photon Bell states that works for continuous electromagnetic waves but not for electromagnetic wave pulses. In addition, only photons emitted in a particular direction are entangled. As a result, only a limited number of photons can be generated. In “Ultrabright source of polarization-entangled photons,” by Kwiat et al., Physical Review A, vol. 60, R773, (1999), Kwiat also describes a source of polarization-entangle photon pairs. However, thin crystals and continuous wave pumps have to be used in order to obtain good entanglement. In “Phase-stable source of polarization-entangled photons using a polarization Sagnac interferometer,” by Taehyun Kim et al., Physical Review A, vol. 73, 012316 (2006) and in “Generation of ultrabright tunable polarization entanglement without spatial, spectral, or temporal constraints,” by Fiorentino et al., Physical Review A, vol. 69, 041801(R) (2004), both Kim and Fiorentino describe an ultrabright parametric down-conversion source of Bell state polarization-entangled photons. However, these polarization-entangled photon sources cannot be used in microscale applications, are expensive to produce, and need periodic adjustments. Physicists have recognized a need for polarization entangled photon sources that are compatible with both continuous wave and pulse pump sources and can be coupled to fiber optic couplers for implementation in microscale devices.